Jacobson Radical Theory
The Jacobson radical of a ring R, denoted by rad R, is the intersection of the maximal left ideals of R. This notion is left-right symmetric; in particular, rad R is an ideal of R. A good way to understand rad R is to think of it as the ideal of elements annihilating all left (resp. right) simple R-modules. The Jacobson radical is also closely tied in with U(R), the group of units of R. In fact, rad R is the largest ideal R such that 1 + R ⊆ U(R).
KeywordsPrime Ideal Maximal Ideal Direct Summand Left Ideal Group Ring
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